In 1976, Lai constructed a nontrivial confidence sequence for the mean $\mu$ of a Gaussian distribution with unknown variance $\sigma$. Curiously, he employed both an improper (right Haar) mixture over $\sigma$ and an improper (flat) mixture over $\mu$. Here, we elaborate carefully on the details of his construction, which use generalized nonintegrable martingales and an extended Ville's inequality. While this does yield a sequential t-test, it does not yield an ``e-process'' (due to the nonintegrability of his martingale). In this paper, we develop two new e-processes and confidence sequences for the same setting: one is a test martingale in a reduced filtration, while the other is an e-process in the canonical data filtration. These are respectively obtained by swapping Lai's flat mixture for a Gaussian mixture, and swapping the right Haar mixture over $\sigma$ with the maximum likelihood estimate under the null, as done in universal inference. We also analyze the width of resulting confidence sequences, which have a curious dependence on the error probability $\alpha$. Numerical experiments are provided along the way to compare and contrast the various approaches.

翻译：1976年，Lai针对方差未知的高斯分布均值$\mu$构建了非平凡置信序列。值得注意的是，他同时采用了$\sigma$上的非正常（右Haar）混合分布和$\mu$上的非正常（平坦）混合分布。本文详细阐述其构造细节——该方法使用了广义不可积鞅与扩展的Ville不等式。虽然该方法确实能导出序贯t检验，但由于其鞅的不可积性，无法生成"e过程"。本文针对同一场景提出了两种新型e过程与置信序列：其一为简化滤子中的检验鞅，其二为标准数据滤子中的e过程。前者通过将Lai的平坦混合替换为高斯混合实现，后者则借鉴通用推断思想，将$\sigma$上的右Haar混合替换为零假设下的极大似然估计。我们进一步分析了所得置信序列的宽度，发现其与误判概率$\alpha$存在特殊关联。文中穿插数值实验以比较各方法的异同。